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Explore Budgeting
Alignment to the Common Core
State Standards for Mathematics
E
xplore Budgeting is a full-year curriculum for
middle or high school students who are participating
in alternate assessments based on alternate
achievement standards. The Explore Budgeting
books—teacher’s manual and student workbook—
use age appropriate content and a highly visual
format for teaching students how to create and use
a budget.
To further assist with alignment of Explore Budgeting
to the Common Core State Standards, the Common
Core Essential Elements and Alternate Achievement
Descriptors for Mathematics (July, 2012), published
by the Wisconsin Department of Public Instruction,
was used. As of July, 2012, thirteen states have
adopted the Common Core Essential Elements for
Mathematics: Iowa, Kansas, Michigan, Mississippi,
Missouri, New Jersey, North Carolina, Oklahoma,
Utah, Virginia, Washington, West Virginia, and
Wisconsin. This document can be accessed for
reference at http://dpi.wi.gov/sped/assmt-ccee.html.
The following document indicates the alignment
of Explore Budgeting to the Common Core State
Standards in Math for grades 6–12. The Common
Core State Standards were written by the National
Governors Association Center for Best Practices in
2010. As of 2013, forty-five states, the District of
Columbia, four territories, and the Department
of Defense Education Activity have adopted the
Common Core State Standards. (The states of Texas,
Alaska, Nebraska, Minnesota, and Virginia, and
the territory of Puerto Rico have not adopted the
standards as of 2013.)
In the following tables, the curriculum’s alignment
to a standard—supported by identified essential
elements and alternate achievement descriptors—
is indicated with the ˜ symbol.
For additional math alignments in the Explore series,
see Explore Math and Explore Math 2.
A
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GEOMETRY
Grade 6
STANDARD
ALIGNMENT
Solve Real-world and Mathematical Problems Involving Area, Surface Area, and Volume 6.G.A.1 – A.4
1.Find the area of right triangles, other triangles, special quadrilaterals, and polygons by
composing into rectangles or decomposing into triangles and other shapes; apply these
techniques in the context of solving real-world and mathematical problems.
2.Find the volume of a right rectangular prism with fractional edge lengths by packing it with
unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the
same as would be found by multiplying the edge lengths of the prism. Apply the formulas
V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge
lengths in the context of solving real-world and mathematical problems.
3.Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates
to find the length of a side joining points with the same first coordinate or the same
second coordinate. Apply these techniques in the context of solving real-world and
mathematical problems.
4.Represent three-dimensional figures using nets made up of rectangles and triangles, and use
the nets to find the surface area of these figures. Apply these techniques in the context of
solving real-world and mathematical problems.
Explore Budgeting • 2 GEOMETRY
Grade 7
STANDARD
ALIGNMENT
Draw, Construct, and Describe Geometrical Figures and Describe the Relationships
Between Them 7.G.A.1 – A.3
1.Solve problems involving scale drawings of geometric figures, including computing actual
lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
2.Draw (freehand, with ruler and protractor, and with technology) geometric shapes with
given conditions. Focus on constructing triangles from three measures of angles or sides,
noticing when the conditions determine a unique triangle, more than one triangle, or
no triangle.
3.Describe the two-dimensional figures that result from slicing three-dimensional figures,
as in plane sections of right rectangular prisms and right rectangular pyramids.
Solve Real-life and Mathematical Problems Involving Angle Measure, Area, Surface Area,
and Volume 7.G.B.4 – B.6
4.Know the formulas for the area and circumference of a circle and use them to solve
problems; give an informal derivation of the relationship between the circumference
and area of a circle.
5.Use facts about supplementary, complementary, vertical, and adjacent angles in a
multi-step problem to write and solve simple equations for an unknown angle in a figure.
6.Solve real-world and mathematical problems involving area, volume and surface area of
two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes,
and right prisms.
Explore Budgeting • 3 GEOMETRY
Grade 8
STANDARD
ALIGNMENT
Understand Congruence and Similarity Using Physical Models, Transparencies,
or Geometry Software 8.G.A.1 – A.5
1. Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
2.Understand that a two-dimensional figure is congruent to another if the second can be
obtained from the first by a sequence of rotations, reflections, and translations; given two
congruent figures, describe a sequence that exhibits the congruence between them.
3.Describe the effect of dilations, translations, rotations, and reflections on two-dimensional
figures using coordinates.
4.Understand that a two-dimensional figure is similar to another if the second can be
obtained from the first by a sequence of rotations, reflections, translations, and dilations;
given two similar two-dimensional figures, describe a sequence that exhibits the similarity
between them.
5.Use informal arguments to establish facts about the angle sum and exterior angle of
triangles, about the angles created when parallel lines are cut by a transversal, and the
angle-angle criterion for similarity of triangles. For example, arrange three copies of the
same triangle so that the sum of the three angles appears to form a line, and give an
argument in terms of transversals why this is so.
Understand and Apply the Pythagorean Theorem 8.G.B.6 – B.8
6. Explain a proof of the Pythagorean Theorem and its converse.
7.Apply the Pythagorean Theorem to determine unknown side lengths in right
triangles in real-world and mathematical problems in two and three dimensions.
8.Apply the Pythagorean Theorem to find the distance between two points in a
coordinate system.
Solve Real-world and Mathematical Problems Involving Volume of Cylinders, Cones, and Spheres 8.G.C.9
9.Know the formulas for the volumes of cones, cylinders, and spheres and use them
to solve real-world and mathematical problems.
Explore Budgeting • 4 RATIOS AND PROPORTIONAL RELATIONSHIPS
Grade 6
STANDARD
ALIGNMENT
Understand Ratio Concepts and Use Ratio Reasoning to Solve Problems 6.RP.A.1 – A.3
1.Understand the concept of a ratio and use ratio language to describe a ratio relationship
between two quantities. For example, “The ratio of wings to beaks in the bird house at the
zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A
received, candidate C received nearly three votes.”
˜
2.Understand the concept of a unit rate a/b associated with a ratio a:b with b  0, and use
rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3
cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid
$75 for 15 hamburgers, which is a rate of $5 per hamburger.” Expectations for unit rates in
this grade are limited to non-complex fractions.
˜
3.Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by
reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams,
or equations.
˜
a.Make tables of equivalent ratios relating quantities with whole-number measurements,
find missing values in the tables, and plot the pairs of values on the coordinate plane.
Use tables to compare ratios.
˜
b.Solve unit rate problems including those involving unit pricing and constant speed. For
example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be
mowed in 35 hours? At what rate were lawns being mowed?
˜
c.Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100
times the quantity); solve problems involving finding the whole, given a part and
the percent.
˜
d.Use ratio reasoning to convert measurement units; manipulate and transform units
appropriately when multiplying or dividing quantities.
˜
Explore Budgeting • 5 RATIOS AND PROPORTIONAL RELATIONSHIPS
Grade 7
STANDARD
ALIGNMENT
Analyze Proportional Relationships and Use Them to Solve Real-world and
Mathematical Problems 7.RP.A.1 – A.3
1.Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and
other quantities measured in like or different units. For example, if a person walks 1/2 mile
in each 1/4 hour, compute the unit rate as the complex fraction 1/2 / 1/4 miles per hour,
equivalently 2 miles per hour.
2.Recognize and represent proportional relationships between quantities.
˜
a.Decide whether two quantities are in a proportional relationship, e.g., by testing for
equivalent ratios in a table or graphing on a coordinate plane and observing whether the
graph is a straight line through the origin.
˜
b.Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams,
and verbal descriptions of proportional relationships.
˜
c.Represent proportional relationships by equations. For example, if total cost t is
proportional to the number n of items purchased at a constant price p, the relationship
between the total cost and the number of items can be expressed as t = pn.
˜
d.Explain what a point (x, y) on the graph of a proportional relationship means in terms of
the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
˜
3.Use proportional relationships to solve multistep ratio and percent problems. Examples:
simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent
increase and decrease, percent error.
˜
Explore Budgeting • 6 THE NUMBER SYSTEM
Grade 6
STANDARD
ALIGNMENT
Apply and Extend Previous Understandings of Multiplication and Division to Divide Fractions
by Fractions 6.NS.A.1
1.Interpret and compute quotients of fractions, and solve word problems involving division
of fractions by fractions, e.g., by using visual fraction models and equations to represent
the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction
model to show the quotient; use the relationship between multiplication and division to
explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) =
ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate
equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a
rectangular strip of land with length 3/4 mi and area 1/2 square mi?
˜
Compute Fluently with Multi-digit Numbers and Find Common Factors and Multiples 6.NS.B.2 – B.4
2. Fluently divide multi-digit numbers using the standard algorithm.
˜
3.Fluently add, subtract, multiply, and divide multi-digit decimals using the standard
algorithm for each operation.
˜
4.Find the greatest common factor of two whole numbers less than or equal to 100 and the
least common multiple of two whole numbers less than or equal to 12. Use the distributive
property to express a sum of two whole numbers 1–100 with a common factor as a multiple
of a sum of two whole numbers with no common factor. For example, express 36 + 8 as
4 (9 + 2).
Apply and Extend Previous Understandings of Numbers to the System of Rational Numbers 6.NS.C.5 – C.8
5.Understand that positive and negative numbers are used together to describe quantities
having opposite directions or values (e.g., temperature above/below zero, elevation above/
below sea level, credits/debits, positive/negative electric charge); use positive and negative
numbers to represent quantities in real-world contexts, explaining the meaning of 0 in
each situation.
˜
6.Understand a rational number as a point on the number line. Extend number line diagrams
and coordinate axes familiar from previous grades to represent points on the line and in the
plane with negative number coordinates.
˜
a.Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on
the number line; recognize that the opposite of the opposite of a number is the number
itself, e.g., –(–3) = 3, and that 0 is its own opposite.
˜
b.Understand signs of numbers in ordered pairs as indicating locations in quadrants of
the coordinate plane; recognize that when two ordered pairs differ only by signs, the
locations of the points are related by reflections across one or both axes.
˜
c.Find and position integers and other rational numbers on a horizontal or vertical
number line diagram; find and position pairs of integers and other rational numbers
on a coordinate plane.
˜
Explore Budgeting • 7 THE NUMBER SYSTEM
Grade 6
STANDARD
ALIGNMENT
7. Understand ordering and absolute value of rational numbers.
˜
a.Interpret statements of inequality as statements about the relative position of two
numbers on a number line diagram. For example, interpret –3 > –7 as a statement
that –3 is located to the right of –7 on a number line oriented from left to right.
˜
b.Write, interpret, and explain statements of order for rational numbers in real-world
contexts. For example, write –3°C > –7°C to express the fact that –3°C is warmer
than –7°C.
˜
c.Understand the absolute value of a rational number as its distance from 0 on the number
line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation. For example, for an account balance of –30 dollars, write |–30| = 30 to
describe the size of the debt in dollars.
˜
d.Distinguish comparisons of absolute value from statements about order. For example,
recognize that an account balance less than –30 dollars represents a debt greater than
30 dollars.
˜
8.Solve real-world and mathematical problems by graphing points in all four quadrants of the
coordinate plane. Include use of coordinates and absolute value to find distances between
points with the same first coordinate or the same second coordinate.
˜
Explore Budgeting • 8 THE NUMBER SYSTEM
Grade 7
STANDARD
ALIGNMENT
Apply and Extend Previous Understandings of Operations with Fractions 7.NS.A.1 – A.3
1.Apply and extend previous understandings of addition and subtraction to add and subtract
rational numbers; represent addition and subtraction on a horizontal or vertical number
line diagram.
˜
a.Describe situations in which opposite quantities combine to make 0. For example, a
hydrogen atom has 0 charge because its two constituents are oppositely charged.
˜
b.Understand p + q as the number located a distance |q| from p, in the positive or negative
direction depending on whether q is positive or negative. Show that a number and its
opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by
describing real-world contexts.
˜
c.Understand subtraction of rational numbers as adding the additive inverse, p – q =
p + (–q). Show that the distance between two rational numbers on the number line is
the absolute value of their difference, and apply this principle in real-world contexts.
˜
d. Apply properties of operations as strategies to add and subtract rational numbers.
˜
2.Apply and extend previous understandings of multiplication and division and of fractions
to multiply and divide rational numbers.
˜
a.Understand that multiplication is extended from fractions to rational numbers by
requiring that operations continue to satisfy the properties of operations, particularly
the distributive property, leading to products such as (–1)(–1) = 1 and the rules for
multiplying signed numbers. Interpret products of rational numbers by describing
real-world contexts.
˜
b.Understand that integers can be divided, provided that the divisor is not zero, and every
quotient of integers (with non-zero divisor) is a rational number. If p and q are integers,
then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing
real-world contexts.
˜
c. Apply properties of operations as strategies to multiply and divide rational numbers.
˜
d.Convert a rational number to a decimal using long division; know that the decimal form
of a rational number terminates in 0s or eventually repeats.
˜
3.Solve real-world and mathematical problems involving the four operations with rational
numbers. Computations with rational numbers extend the rules for manipulating fractions
to complex fractions.
˜
Explore Budgeting • 9 THE NUMBER SYSTEM
Grade 8
STANDARD
ALIGNMENT
Know That There Are Numbers That Are Not Rational, and Approximate Them by
Rational Numbers 8.NS.A.1 – A.2
1.Know that numbers that are not rational are called irrational. Understand informally
that every number has a decimal expansion; for rational numbers show that the decimal
expansion repeats eventually, and convert a decimal expansion which repeats eventually
into a rational number.
˜
2.Use rational approximations of irrational numbers to compare the size of irrational
numbers, locate them approximately on a number line diagram, and estimate the value
of expressions (e.g., 2). For example, by truncating the decimal expansion of √2, show
that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on
to get better approximations.
˜
Explore Budgeting • 10 EXPRESSIONS AND EQUATIONS
Grade 6
STANDARD
ALIGNMENT
Apply and Extend Previous Understandings of Arithmetic to Algebraic Expressions 6.EE.A.1 – A.4
1. Write and evaluate numerical expressions involving whole-number exponents.
˜
2. Write, read, and evaluate expressions in which letters stand for numbers.
˜
a.Write expressions that record operations with numbers and with letters standing for
numbers. For example, express the calculation “Subtract y from 5” as 5 – y.
˜
b.Identify parts of an expression using mathematical terms (sum, term, product, factor,
quotient, coefficient); view one or more parts of an expression as a single entity. For
example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7)
as both a single entity and a sum of two terms.
˜
c.Evaluate expressions at specific values of their variables. Include expressions that arise
from formulas used in real-world problems. Perform arithmetic operations, including
those involving whole-number exponents, in the conventional order when there are no
parentheses to specify a particular order (Order of Operations). For example, use the
formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of
length s = 1/2.
˜
3.Apply the properties of operations to generate equivalent expressions. For example, apply
the distributive property to the expression 3 (2 + x) to produce the equivalent expression
6 + 3x; apply the distributive property to the expression 24x + 18y to produce the
equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce
the equivalent expression 3y.
˜
4.Identify when two expressions are equivalent (i.e., when the two expressions name the same
number regardless of which value is substituted into them). For example, the expressions
y + y + y and 3y are equivalent because they name the same number regardless of which
number y stands for.
˜
Reason About and Solve One-variable Equations and Inequalities 6.EE.B.5 – B.8
5.Understand solving an equation or inequality as a process of answering a question: which
values from a specified set, if any, make the equation or inequality true? Use substitution to
determine whether a given number in a specified set makes an equation or inequality true.
˜
6.Use variables to represent numbers and write expressions when solving a real-world or
mathematical problem; understand that a variable can represent an unknown number, or,
depending on the purpose at hand, any number in a specified set.
˜
7.Solve real-world and mathematical problems by writing and solving equations of the form
x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
˜
8.Write an inequality of the form x > c or x < c to represent a constraint or condition in a realworld or mathematical problem. Recognize that inequalities of the form x > c or x < c have
infinitely many solutions; represent solutions of such inequalities on number line diagrams.
˜
Explore Budgeting • 11 EXPRESSIONS AND EQUATIONS
Grade 6
STANDARD
ALIGNMENT
Represent and Analyze Quantitative Relationships Between Dependent and Independent Variables 6.EE.C.9
9.Use variables to represent two quantities in a real-world problem that change in relationship
to one another; write an equation to express one quantity, thought of as the dependent
variable, in terms of the other quantity, thought of as the independent variable. Analyze
the relationship between the dependent and independent variables using graphs and tables,
and relate these to the equation. For example, in a problem involving motion at constant
speed, list and graph ordered pairs of distances and times, and write the equation d = 65t
to represent the relationship between distance and time.
Explore Budgeting • 12 EXPRESSIONS AND EQUATIONS
Grade 7
STANDARD
ALIGNMENT
Use Properties of Operations to Generate Equivalent Expressions 7.EE.A.1 – A.2
1.Apply properties of operations as strategies to add, subtract, factor, and expand linear
expressions with rational coefficients.
˜
2.Understand that rewriting an expression in different forms in a problem context can shed
light on the problem and how the quantities in it are related. For example, a + 0.05a =
1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
˜
Solve Real-life and Mathematical Problems Using Numerical and Algebraic Expressions
and Equations 7.EE.B.3 – B.4
3.Solve multi-step real-life and mathematical problems posed with positive and negative
rational numbers in any form (whole numbers, fractions, and decimals), using tools
strategically. Apply properties of operations to calculate with numbers in any form; convert
between forms as appropriate; and assess the reasonableness of answers using mental
computation and estimation strategies. For example: If a woman making $25 an hour gets a
10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary
of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is
27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this
estimate can be used as a check on the exact computation.
˜
4.Use variables to represent quantities in a real-world or mathematical problem, and construct
simple equations and inequalities to solve problems by reasoning about the quantities.
˜
a.Solve word problems leading to equations of the form px + q = r and p(x + q) = r,
where p, q, and r are specific rational numbers. Solve equations of these forms fluently.
Compare an algebraic solution to an arithmetic solution, identifying the sequence of the
operations used in each approach. For example, the perimeter of a rectangle is 54 cm.
Its length is 6 cm. What is its width?
˜
b.Solve word problems leading to inequalities of the form px + q > r or px + q < r, where
p, q, and r are specific rational numbers. Graph the solution set of the inequality and
interpret it in the context of the problem. For example: As a salesperson, you are paid
$50 per week plus $3 per sale. This week you want your pay to be at least $100. Write
an inequality for the number of sales you need to make, and describe the solutions.
˜
Explore Budgeting • 13 EXPRESSIONS AND EQUATIONS
Grade 8
STANDARD
ALIGNMENT
Expressions and Equations Work with Radicals and Integer Exponents 8.EE.A.1 – A.4
1.Know and apply the properties of integer exponents to generate equivalent numerical
expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.
˜
2.Use square root and cube root symbols to represent solutions to equations of the form
x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small
perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
˜
3.Use numbers expressed in the form of a single digit times an integer power of 10 to
estimate very large or very small quantities, and to express how many times as much one
is than the other. For example, estimate the population of the United States as 3 times 108
and the population of the world as 7 times 109, and determine that the world population
is more than 20 times larger.
˜
4.Perform operations with numbers expressed in scientific notation, including problems
where both decimal and scientific notation are used. Use scientific notation and choose
units of appropriate size for measurements of very large or very small quantities (e.g., use
millimeters per year for seafloor spreading). Interpret scientific notation that has been
generated by technology.
˜
Understand the Connections Between Proportional Relationships, Lines, and Linear Equations 8.EE.B.5 – B.6
5.Graph proportional relationships, interpreting the unit rate as the slope of the graph.
Compare two different proportional relationships represented in different ways. For
example, compare a distance-time graph to a distance-time equation to determine
which of two moving objects has greater speed.
˜
6.Use similar triangles to explain why the slope m is the same between any two distinct points
on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through
the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
˜
Analyze and Solve Linear Equations and Pairs of Simultaneous Linear Equations 8.EE.C.7 – C.8
7. Solve linear equations in one variable.
a.Give examples of linear equations in one variable with one solution, infinitely many
solutions, or no solutions. Show which of these possibilities is the case by successively
transforming the given equation into simpler forms, until an equivalent equation of the
form x = a, a = a, or a = b results (where a and b are different numbers).
˜
b.Solve linear equations with rational number coefficients, including equations whose
solutions require expanding expressions using the distributive property and collecting
like terms.
˜
Explore Budgeting • 14 EXPRESSIONS AND EQUATIONS
Grade 8
STANDARD
ALIGNMENT
8. Analyze and solve pairs of simultaneous linear equations.
a.Understand that solutions to a system of two linear equations in two variables
correspond to points of intersection of their graphs, because points of intersection
satisfy both equations simultaneously.
b.Solve systems of two linear equations in two variables algebraically, and estimate
solutions by graphing the equations. Solve simple cases by inspection. For example,
3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously
be 5 and 6.
c.Solve real-world and mathematical problems leading to two linear equations in two
variables. For example, given coordinates for two pairs of points, determine whether
the line through the first pair of points intersects the line through the second pair.
Explore Budgeting • 15 STATISTICS AND PROBABILITY
Grade 6
STANDARD
ALIGNMENT
Develop Understanding of Statistical Variability 6.SP.A.1 – A.3
1.Recognize a statistical question as one that anticipates variability in the data related to the
question and accounts for it in the answers. For example, “How old am I?” is not a statistical
question, but “How old are the students in my school?” is a statistical question because one
anticipates variability in students’ ages.
2.Understand that a set of data collected to answer a statistical question has a distribution
which can be described by its center, spread, and overall shape.
3.Recognize that a measure of center for a numerical data set summarizes all of its values
with a single number, while a measure of variation describes how its values vary with a
single number.
Summarize and Describe Distributions 6.SP.B.4 – B.5
4.Display numerical data in plots on a number line, including dot plots, histograms, and
box plots.
5.Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations.
b.Describing the nature of the attribute under investigation, including how it was
measured and its units of measurement.
c.Giving quantitative measures of center (median and/or mean) and variability
(interquartile range and/or mean absolute deviation), as well as describing any
overall pattern and any striking deviations from the overall pattern with reference
to the context in which the data were gathered.
d.Relating the choice of measures of center and variability to the shape of the data
distribution and the context in which the data were gathered.
Explore Budgeting • 16 STATISTICS AND PROBABILITY
Grade 7
STANDARD
ALIGNMENT
Use Random Sampling to Draw Inferences About a Population 7.SP.A.1 – A.2
1.Understand that statistics can be used to gain information about a population by examining
a sample of the population; generalizations about a population from a sample are valid only
if the sample is representative of that population. Understand that random sampling tends
to produce representative samples and support valid inferences.
2.Use data from a random sample to draw inferences about a population with an unknown
characteristic of interest. Generate multiple samples (or simulated samples) of the same
size to gauge the variation in estimates or predictions. For example, estimate the mean
word length in a book by randomly sampling words from the book; predict the winner
of a school election based on randomly sampled survey data. Gauge how far off the
estimate or prediction might be.
Draw Informal Comparative Inferences About Two Populations 7.SP.B.3 – B.4
3.Informally assess the degree of visual overlap of two numerical data distributions with similar
variabilities, measuring the difference between the centers by expressing it as a multiple of a
measure of variability. For example, the mean height of players on the basketball team is 10
cm greater than the mean height of players on the soccer team, about twice the variability
(mean absolute deviation) on either team; on a dot plot, the separation between the two
distributions of heights is noticeable.
4.Use measures of center and measures of variability for numerical data from random samples
to draw informal comparative inferences about two populations. For example, decide
whether the words in a chapter of a seventh-grade science book are generally longer than
the words in a chapter of a fourth-grade science book.
Investigate Chance Processes and Develop, Use, and Evaluate Probability Models 7.SP.C.5 – C.8
5.Understand that the probability of a chance event is a number between 0 and 1 that
expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood.
A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event
that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
6.Approximate the probability of a chance event by collecting data on the chance process
that produces it and observing its long-run relative frequency, and predict the approximate
relative frequency given the probability. For example, when rolling a number cube 600
times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly
200 times.
7.Develop a probability model and use it to find probabilities of events. Compare probabilities
from a model to observed frequencies; if the agreement is not good, explain possible
sources of the discrepancy.
a.Develop a uniform probability model by assigning equal probability to all outcomes, and
use the model to determine probabilities of events. For example, if a student is selected
at random from a class, find the probability that Jane will be selected and the probability
that a girl will be selected.
b.Develop a probability model (which may not be uniform) by observing frequencies in
data generated from a chance process. For example, find the approximate probability
that a spinning penny will land heads up or that a tossed paper cup will land open-end
down. Do the outcomes for the spinning penny appear to be equally likely based on the
observed frequencies?
Explore Budgeting • 17 STATISTICS AND PROBABILITY
Grade 7
STANDARD
ALIGNMENT
8.Find probabilities of compound events using organized lists, tables, tree diagrams,
and simulation.
a.Understand that, just as with simple events, the probability of a compound event is
the fraction of outcomes in the sample space for which the compound event occurs.
b.Represent sample spaces for compound events using methods such as organized lists,
tables and tree diagrams. For an event described in everyday language (e.g., “rolling
double sixes”), identify the outcomes in the sample space which compose the event.
c.Design and use a simulation to generate frequencies for compound events. For example,
use random digits as a simulation tool to approximate the answer to the question:
If 40% of donors have type A blood, what is the probability that it will take at least
4 donors to find one with type A blood?
Explore Budgeting • 18 STATISTICS AND PROBABILITY
Grade 8
STANDARD
ALIGNMENT
Investigate Patterns of Association in Bivariate Data 8.SP.A.1 – A.4
1.Construct and interpret scatter plots for bivariate measurement data to investigate patterns
of association between two quantities. Describe patterns such as clustering, outliers, positive
or negative association, linear association, and nonlinear association.
2.Know that straight lines are widely used to model relationships between two quantitative
variables. For scatter plots that suggest a linear association, informally fit a straight line, and
informally assess the model fit by judging the closeness of the data points to the line.
3.Use the equation of a linear model to solve problems in the context of bivariate
measurement data, interpreting the slope and intercept. For example, in a linear model
for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional
hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
4.Understand that patterns of association can also be seen in bivariate categorical data by
displaying frequencies and relative frequencies in a two-way table. Construct and interpret
a two-way table summarizing data on two categorical variables collected from the same
subjects. Use relative frequencies calculated for rows or columns to describe possible
association between the two variables. For example, collect data from students in your
class on whether or not they have a curfew on school nights and whether or not they
have assigned chores at home. Is there evidence that those who have a curfew also tend
to have chores?
Explore Budgeting • 19 FUNCTIONS
Grade 8
STANDARD
ALIGNMENT
Define, Evaluate, and Compare Functions 8.F.A.1 – A.3
1.Understand that a function is a rule that assigns to each input exactly one output. The graph
of a function is the set of ordered pairs consisting of an input and the corresponding output.
2.Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). For example, given a linear
function represented by a table of values and a linear function represented by an algebraic
expression, determine which function has the greater rate of change.
3.Interpret the equation y = mx + b as defining a linear function, whose graph is a straight
line; give examples of functions that are not linear. For example, the function A = s2 giving
the area of a square as a function of its side length is not linear because its graph contains
the points (1,1), (2,4) and (3,9), which are not on a straight line.
Use Functions to Model Relationships Between Quantities 8.F.B.4 – B.5
4.Construct a function to model a linear relationship between two quantities. Determine the
rate of change and initial value of the function from a description of a relationship or from
two (x, y) values, including reading these from a table or from a graph. Interpret the rate of
change and initial value of a linear function in terms of the situation it models, and in terms
of its graph or a table of values.
˜
5.Describe qualitatively the functional relationship between two quantities by analyzing a
graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a
graph that exhibits the qualitative features of a function that has been described verbally.
˜
Explore Budgeting • 20 HIGH SCHOOL: NUMBER AND QUANTITY
The Real Number System
STANDARD
ALIGNMENT
Extend the Properties of Exponents to Rational Exponents HSN-RN.A.1 – A.2
1.Explain how the definition of the meaning of rational exponents follows from extending the
properties of integer exponents to those values, allowing for a notation for radicals in terms
of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want
(5 1/3)3 = 5 1/3)3 to hold, so (5 1/3)3 must equal 5.
2.Rewrite expressions involving radicals and rational exponents using the properties
of exponents.
Use Properties of Rational and Irrational Numbers HSN-RN.B.3
3.Explain why the sum or product of two rational numbers is rational; that the sum of a
rational number and an irrational number is irrational; and that the product of a nonzero
rational number and an irrational number is irrational.
Explore Budgeting • 21 HIGH SCHOOL: NUMBER AND QUANTITY
Quantities
STANDARD
ALIGNMENT
Reason Quantitatively and Use Units to Solve Problems HSN-Q.A.1 – A.3
1.Use units as a way to understand problems and to guide the solution of multi-step
problems; choose and interpret units consistently in formulas; choose and interpret
the scale and the origin in graphs and data displays.
˜
2. Define appropriate quantities for the purpose of descriptive modeling.
˜
3.Choose a level of accuracy appropriate to limitations on measurement when
reporting quantities.
˜
Explore Budgeting • 22 HIGH SCHOOL: NUMBER AND QUANTITY
The Complex Number System
STANDARD
ALIGNMENT
Perform Arithmetic Operations with Complex Numbers HSN-CN.A.1 – A.3
1.Know there is a complex number i such that i2 = –1, and every complex number has the
form a + bi with a and b real.
2.Use the relation i2 = –1 and the commutative, associative, and distributive properties to
add, subtract, and multiply complex numbers.
˜
3.Find the conjugate of a complex number; use conjugates to find moduli and quotients
of complex numbers.
Represent Complex Numbers and Their Operations on the Complex Plane HSN-CN.B.4 – B.6
4.Represent complex numbers on the complex plane in rectangular and polar form
(including real and imaginary numbers), and explain why the rectangular and polar
forms of a given complex number represent the same number.
5.Represent addition, subtraction, multiplication, and conjugation of complex numbers
geometrically on the complex plane; use properties of this representation for computation.
For example, (–1 + √3 i)3 = 8 because (–1 + √3 i) has modulus 2 and argument 120°.
6.Calculate the distance between numbers in the complex plane as the modulus of the
difference, and the midpoint of a segment as the average of the numbers at its endpoints.
Use Complex Numbers in Polynomial Identities and Equations HSN-CN.C.7 – C.9
7.Solve quadratic equations with real coefficients that have complex solutions.
8.Extend polynomial identities to the complex numbers. For example, rewrite
x2 + 4 as (x + 2i)(x – 2i).
9.Know the Fundamental Theorem of Algebra; show that it is true for
quadratic polynomials.
Explore Budgeting • 23 HIGH SCHOOL: NUMBER AND QUANTITY
Vector and Matric Quantities
STANDARD
ALIGNMENT
Represent and Model with Vector Quantites HSN-VM.A.1 – A.3
1.Recognize vector quantities as having both magnitude and direction. Represent vector
quantities by directed line segments, and use appropriate symbols for vectors and their
magnitudes (e.g., v, |v|, ||v||, v).
2.Find the components of a vector by subtracting the coordinates of an initial point from
the coordinates of a terminal point.
3.Solve problems involving velocity and other quantities that can be represented
by vectors.
Perform Operations on Vectors HSN-VM.B.4 – B.5
4. Add and subtract vectors.
a.Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand
that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
b.Given two vectors in magnitude and direction form, determine the magnitude and
direction of their sum.
c.Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w,
with the same magnitude as w and pointing in the opposite direction. Represent vector
subtraction graphically by connecting the tips in the appropriate order, and perform
vector subtraction component-wise.
5.Multiply a vector by a scalar.
a.Represent scalar multiplication graphically by scaling vectors and possibly reversing their
direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).
b.Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction
of cv knowing that when |c|v 0, the direction of cv is either along v (for c > 0) or against
v (for c < 0).
Perform Operations on Matrices and Use Matrices in Applications HSN-VM.C.6 – C.12
6.Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence
relationships in a network.
7.Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs
in a game are doubled.
8.Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs
in a game are doubled.
9.Understand that, unlike multiplication of numbers, matrix multiplication for square matrices
is not a commutative operation, but still satisfies the associative and distributive properties.
10.Understand that the zero and identity matrices play a role in matrix addition and
multiplication similar to the role of 0 and 1 in the real numbers. The determinant of
a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
11.Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions
to produce another vector. Work with matrices as transformations of vectors.
12.Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value
of the determinant in terms of area.
Explore Budgeting • 24 HIGH SCHOOL: ALGEBRA
Seeing Structure in Expressions
STANDARD
ALIGNMENT
Interpret the Structure of Expressions HSA-SSE.A.1 – A.2
1. Interpret expressions that represent a quantity in terms of its context.
˜
a. Interpret parts of an expression, such as terms, factors, and coefficients.
˜
b.Interpret complicated expressions by viewing one or more of their parts as a single
entity. For example, interpret P(1 + r)n as the product of P and a factor not depending
on P.
˜
2.Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4
as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as
(x2 – y2)(x2 + y2).
Write Expressions in Equivalent Forms to Solve Problems HSA-SSE.B.3 – B.4
3.Choose and produce an equivalent form of an expression to reveal and explain properties
of the quantity represented by the expression.
˜
a. Factor a quadratic expression to reveal the zeros of the function it defines.
˜
b.Complete the square in a quadratic expression to reveal the maximum or minimum
value of the function it defines.
˜
c.Use the properties of exponents to transform expressions for exponential functions.
For example the expression 1.15t can be rewritten as (1.151/12)12t  1.01212t to reveal
the approximate equivalent monthly interest rate if the annual rate is 15%.
˜
4.Derive the formula for the sum of a finite geometric series (when the common ratio is not
1), and use the formula to solve problems. For example, calculate mortgage payments.
Explore Budgeting • 25 HIGH SCHOOL: ALGEBRA
Arithmetic with Polynomials and Rational Expressions
STANDARD
ALIGNMENT
Perform Arithmetic Operations on Polynomials HSA-APR.A.1
1.Understand that polynomials form a system analogous to the integers, namely, they are
closed under the operations of addition, subtraction, and multiplication; add, subtract, and
multiply polynomials.
Understand the Relationship Between Zeros and Factors of Polynomials HSA-APR.B.2 – B.3
2.Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the
remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
3.Identify zeros of polynomials when suitable factorizations are available, and use the zeros
to construct a rough graph of the function defined by the polynomial.
Use Polynomial Identities to Solve Problems HSA-APR.C.4 – C.5
4.Prove polynomial identities and use them to describe numerical relationships. For
example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate
Pythagorean triples.
5.Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y
for a positive integer n, where x and y are any numbers, with coefficients determined for
example by Pascal’s Triangle.
Rewrite Rational Expressions HSA-APR.D.6 – D.7
6.Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x),
where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree
of b(x), using inspection, long division, or, for the more complicated examples,
a computer algebra system.
7.Understand that rational expressions form a system analogous to the rational numbers,
closed under addition, subtraction, multiplication, and division by a nonzero rational
expression; add, subtract, multiply, and divide rational expressions.
Explore Budgeting • 26 HIGH SCHOOL: ALGEBRA
Creating Equations
STANDARD
ALIGNMENT
Create Equations that Describe Numbers or Relationships HSA-CED.A.1 – A.4
1.Create equations and inequalities in one variable and use them to solve problems.
Include equations arising from linear and quadratic functions, and simple rational and
exponential functions.
˜
2.Create equations in two or more variables to represent relationships between quantities;
graph equations on coordinate axes with labels and scales.★
˜
3.Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or nonviable options in a modeling context.
For example, represent inequalities describing nutritional and cost constraints on
combinations of different foods.
˜
4.Rearrange formulas to highlight a quantity of interest, using the same reasoning as in
solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
˜
Explore Budgeting • 27 HIGH SCHOOL: ALGEBRA
Reasoning with Equations and Inequalities
STANDARD
ALIGNMENT
Understand Solving Equations as a Process of Reasoning and Explain the Reasoning HSA-REI.A.1 – A.2
1.Explain each step in solving a simple equation as following from the equality of numbers
asserted at the previous step, starting from the assumption that the original equation has
a solution. Construct a viable argument to justify a solution method.
2.Solve simple rational and radical equations in one variable, and give examples showing how
extraneous solutions may arise.
Solve Equations and Inequalities in One Variable HSA-REI.B.3 – B.4
3.Solve linear equations and inequalities in one variable, including equations with coefficients
represented by letters.
4.Solve quadratic equations in one variable.
a.Use the method of completing the square to transform any quadratic equation in x into
an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic
formula from this form.
b.Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots,
completing the square, the quadratic formula and factoring, as appropriate to the initial
form of the equation. Recognize when the quadratic formula gives complex solutions
and write them as a ± bi for real numbers a and b.
Solve Systems of Equations HSA-REI.C.5 – C.9
5.Prove that, given a system of two equations in two variables, replacing one equation by the
sum of that equation and a multiple of the other produces a system with the same solutions.
6.Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on
pairs of linear equations in two variables.
7.Solve a simple system consisting of a linear equation and a quadratic equation in two
variables algebraically and graphically. For example, find the points of intersection between
the line y = –3x and the circle x2 + y2 = 3.
8.Represent a system of linear equations as a single matrix equation in a vector variable.
9.Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using
technology for matrices of dimension 3 × 3 or greater).
Represent and Solve Equations and Inequalities Graphically HSA-REI.D.10 – D.12
10.Understand that the graph of an equation in two variables is the set of all its solutions
plotted in the coordinate plane, often forming a curve (which could be a line).
˜
11.Explain why the x-coordinates of the points where the graphs of the equations y = f(x)
and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions
approximately, e.g., using technology to graph the functions, make tables of values, or
find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial,
rational, absolute value, exponential, and logarithmic functions. ★
˜
12.Graph the solutions to a linear inequality in two variables as a half-plane (excluding the
boundary in the case of a strict inequality), and graph the solution set to a system of linear
inequalities in two variables as the intersection of the corresponding half-planes.
˜
Explore Budgeting • 28 HIGH SCHOOL: FUNCTIONS
Interpreting Functions
STANDARD
ALIGNMENT
Understand the Concept of a Function and Use Function Notation HSF-IF.A.1 – A.3
1.Understand that a function from one set (called the domain) to another set (called the
range) assigns to each element of the domain exactly one element of the range. If f is a
function and x is an element of its domain, then f(x) denotes the output of f corresponding
to the input x. The graph of f is the graph of the equation y = f(x).
˜
2.Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
˜
3.Recognize that sequences are functions, sometimes defined recursively, whose domain
is a subset of the integers. For example, the Fibonacci sequence is defined recursively by
f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
˜
Interpret Functions That Arise in Applications in Terms of the Context HSF-IF.B.4 – B.6
4.For a function that models a relationship between two quantities, interpret key features of
graphs and tables in terms of the quantities, and sketch graphs showing key features given
a verbal description of the relationship. Key features include: intercepts; intervals where the
function is increasing, decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.
˜
5.Relate the domain of a function to its graph and, where applicable, to the quantitative
relationship it describes. For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an
appropriate domain for the function.
6.Calculate and interpret the average rate of change of a function (presented symbolically
or as a table) over a specified interval. Estimate the rate of change from a graph.
Analyze Functions Using Different Representations HSF-IF.C.7 – C.9
7.Graph functions expressed symbolically and show key features of the graph, by hand in
simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
b.Graph square root, cube root, and piecewise-defined functions, including step functions
and absolute value functions.
c.Graph polynomial functions, identifying zeros when suitable factorizations are available,
and showing end behavior.
d.Graph rational functions, identifying zeros and asymptotes when suitable factorizations
are available, and showing end behavior.
e.Graph exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showing period, midline, and amplitude.
Explore Budgeting • 29 HIGH SCHOOL: FUNCTIONS
Interpreting Functions
STANDARD
ALIGNMENT
8.Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the function.
a.Use the process of factoring and completing the square in a quadratic function to
show zeros, extreme values, and symmetry of the graph, and interpret these in terms
of a context.
b.Use the properties of exponents to interpret expressions for exponential functions. For
example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y =
(1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
9.Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). For example, given a graph
of one quadratic function and an algebraic expression for another, say which has the
larger maximum.
Explore Budgeting • 30 HIGH SCHOOL: FUNCTIONS
Building Functions
STANDARD
ALIGNMENT
Build a Function That Models a Relationship Between Two Quantities HSF-BF.A.1 – A.2
1. Write a function that describes a relationship between two quantities.
a.Determine an explicit expression, a recursive process, or steps for calculation from
a context.
b.Combine standard function types using arithmetic operations. For example, build a
function that models the temperature of a cooling body by adding a constant function
to a decaying exponential, and relate these functions to the model.
c.Compose functions. For example, if T(y) is the temperature in the atmosphere as a
function of height, and h(t) is the height of a weather balloon as a function of time, then
T(h(t)) is the temperature at the location of the weather balloon as a function of time.
2.Write arithmetic and geometric sequences both recursively and with an explicit formula, use
them to model situations, and translate between the two forms.
Build New Functions from Existing Functions HSF-BF.B.3 – B.5
3.Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for
specific values of k (both positive and negative); find the value of k given the graphs.
Experiment with cases and illustrate an explanation of the effects on the graph using
technology. Include recognizing even and odd functions from their graphs and algebraic
expressions for them.
4. Find inverse functions.
a.Solve an equation of the form f(x) = c for a simple function f that has an inverse and
write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for
x  1.
b.Verify by composition that one function is the inverse of another.
c.Read values of an inverse function from a graph or a table, given that the function
has an inverse.
d.Produce an invertible function from a non-invertible function by restricting
the domain.
5.Understand the inverse relationship between exponents and logarithms and use this
relationship to solve problems involving logarithms and exponents.
Explore Budgeting • 31 HIGH SCHOOL: FUNCTIONS
Linear, Quadratic, and Exponential Models
STANDARD
ALIGNMENT
Construct and Compare Linear, Quadratic, and Exponential Models and Solve Problems HSF-LE.A.1 – A.4
1.Distinguish between situations that can be modeled with linear functions and with
exponential functions.
˜
a.Prove that linear functions grow by equal differences over equal intervals, and that
exponential functions grow by equal factors over equal intervals.
˜
b.Recognize situations in which one quantity changes at a constant rate per unit interval
relative to another.
˜
c.Recognize situations in which a quantity grows or decays by a constant percent rate per
unit interval relative to another.
˜
2.Construct linear and exponential functions, including arithmetic and geometric sequences,
given a graph, a description of a relationship, or two input-output pairs (include reading
these from a table).
˜
3.Observe using graphs and tables that a quantity increasing exponentially eventually exceeds
a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
˜
4.For exponential models, express as a logarithm the solution to abct = d where a, c, and d are
numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
˜
Interpret Expressions for Functions in Terms of the Situation They Model HSF-LE.B.5
5. Interpret the parameters in a linear or exponential function in terms of a context.
Explore Budgeting • 32 HIGH SCHOOL: FUNCTIONS
Trigonometric Functions
STANDARD
ALIGNMENT
Extend the Domain of Trigonometric Functions Using the Unit Circle HSF-TF.A.1 – A.4
1.Understand radian measure of an angle as the length of the arc on the unit circle subtended
by the angle.
2.Explain how the unit circle in the coordinate plane enables the extension of trigonometric
functions to all real numbers, interpreted as radian measures of angles traversed
counterclockwise around the unit circle.
3.Use special triangles to determine geometrically the values of sine, cosine, tangent
for /3, /4 and /6, and use the unit circle to express the values of sine, cosine, and
tangent for x,  + x, and 2 – x in terms of their values for x, where x is any real number.
4.Use the unit circle to explain symmetry (odd and even) and periodicity of
trigonometric functions.
Model Periodic Phenomena with Trigonometric Functions HSF-TF.B.5 – B.7
5.Choose trigonometric functions to model periodic phenomena with specified amplitude,
frequency, and midline.
6.Understand that restricting a trigonometric function to a domain on which it is always
increasing or always decreasing allows its inverse to be constructed.
7.Use inverse functions to solve trigonometric equations that arise in modeling contexts;
evaluate the solutions using technology, and interpret them in terms of the context.
Prove and Apply Trigonometric Identities HSF-TF.C.8 – C.9
8.Prove the Pythagorean identity sin2() + cos2() = 1 and use it to find sin(), cos(),
or tan() given sin(), cos(), or tan() and the quadrant of the angle.
9.Prove the addition and subtraction formulas for sine, cosine, and tangent and use them
to solve problems.
Explore Budgeting • 33 HIGH SCHOOL: GEOMETRY
Congruence
STANDARD
ALIGNMENT
Experiment with Transformations in the Plane HSG-CO.A.1 – A.5
1.Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment,
based on the undefined notions of point, line, distance along a line, and distance around a
circular arc.
2.Represent transformations in the plane using, e.g., transparencies and geometry software;
describe transformations as functions that take points in the plane as inputs and give other
points as outputs. Compare transformations that preserve distance and angle to those that
do not (e.g., translation versus horizontal stretch).
3.Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and
reflections that carry it onto itself.
4.Develop definitions of rotations, reflections, and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments.
5.Given a geometric figure and a rotation, reflection, or translation, draw the transformed
figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence
of transformations that will carry a given figure onto another.
Understand Congruence in Terms of Rigid Motions HSG-CO.B.6 – B.8
6.Use geometric descriptions of rigid motions to transform figures and to predict the effect
of a given rigid motion on a given figure; given two figures, use the definition of congruence
in terms of rigid motions to decide if they are congruent.
7.Use the definition of congruence in terms of rigid motions to show that two triangles
are congruent if and only if corresponding pairs of sides and corresponding pairs of angles
are congruent.
8.Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the
definition of congruence in terms of rigid motions.
Prove Geometric Theorems HSG-CO.C.9 – C.11
9.Prove theorems about lines and angles. Theorems include: vertical angles are congruent;
when a transversal crosses parallel lines, alternate interior angles are congruent and
corresponding angles are congruent; points on a perpendicular bisector of a line segment
are exactly those equidistant from the segment’s endpoints.
10.Prove theorems about triangles. Theorems include: measures of interior angles of a triangle
sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints
of two sides of a triangle is parallel to the third side and half the length; the medians of a
triangle meet at a point.
11.Prove theorems about parallelograms. Theorems include: opposite sides are congruent,
opposite angles are congruent, the diagonals of a parallelogram bisect each other, and
conversely, rectangles are parallelograms with congruent diagonals.
Explore Budgeting • 34 HIGH SCHOOL: GEOMETRY
Congruence
STANDARD
ALIGNMENT
Make Geometric Constructions HSG-CO.D.12 – D.13
12.Make formal geometric constructions with a variety of tools and methods (compass and
straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing
perpendicular lines, including the perpendicular bisector of a line segment; and constructing
a line parallel to a given line through a point not on the line.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Explore Budgeting • 35 HIGH SCHOOL: GEOMETRY
Similarity, Right Triangles, and Trigonometry
STANDARD
ALIGNMENT
Understand Similarity in Terms of Similarity Transformations HSG-SRT.A.1 – A.3
1. Verify experimentally the properties of dilations given by a center and a scale factor:
a.A dilation takes a line not passing through the center of the dilation to a parallel line,
and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
2.Given two figures, use the definition of similarity in terms of similarity transformations to
decide if they are similar; explain using similarity transformations the meaning of similarity
for triangles as the equality of all corresponding pairs of angles and the proportionality of
all corresponding pairs of sides.
3.Use the properties of similarity transformations to establish the AA criterion for two triangles
to be similar.
Prove Theorems Involving Similarity HSG-SRT.B.4 – B.5
4.Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle
divides the other two proportionally, and conversely; the Pythagorean Theorem proved
using triangle similarity.
5.Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
Define Trigonometric Ratios and Solve Problems Involving Right Triangles HSG-SRT.C.6 – C.8
6.Understand that by similarity, side ratios in right triangles are properties of the angles in the
triangle, leading to definitions of trigonometric ratios for acute angles.
7.Explain and use the relationship between the sine and cosine of complementary angles.
8.Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems.
Apply Trigonometry to General Triangles HSG-SRT.D.9 – D.11
9.Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line
from a vertex perpendicular to the opposite side.
10.Prove the Laws of Sines and Cosines and use them to solve problems.
11.Understand and apply the Law of Sines and the Law of Cosines to find unknown
measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
Explore Budgeting • 36 HIGH SCHOOL: GEOMETRY
Circles
STANDARD
ALIGNMENT
Understand and Apply Theorems About Circles HSG-C.A.1 – A.4
1. Prove that all circles are similar.
2.Identify and describe relationships among inscribed angles, radii, and chords. Include the
relationship between central, inscribed, and circumscribed angles; inscribed angles on a
diameter are right angles; the radius of a circle is perpendicular to the tangent where the
radius intersects the circle.
3.Construct the inscribed and circumscribed circles of a triangle, and prove properties of
angles for a quadrilateral inscribed in a circle.
4.Construct a tangent line from a point outside a given circle to the circle.
Find Arc Lengths and Areas of Sectors of Circles HSG-C.B.5
5.Derive using similarity the fact that the length of the arc intercepted by an angle is
proportional to the radius, and define the radian measure of the angle as the constant
of proportionality; derive the formula for the area of a sector.
Explore Budgeting • 37 HIGH SCHOOL: GEOMETRY
Expressing Geometric Properties with Equations
STANDARD
ALIGNMENT
Translate Between the Geometric Description and the Equation for a Conic Section HSG-GPE.A.1 – A.3
1.Derive the equation of a circle of given center and radius using the Pythagorean Theorem;
complete the square to find the center and radius of a circle given by an equation.
2.Derive the equation of a parabola given a focus and directrix.
3.Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or
difference of distances from the foci is constant.
Use Coordinates to Prove Simple Geometric Theorems Algebraically HSG-GPE.B.4 – B.7
4.Use coordinates to prove simple geometric theorems algebraically. For example, prove or
disprove that a figure defined by four given points in the coordinate plane is a rectangle;
prove or disprove that the point (1, √3) lies on the circle centered at the origin and
containing the point (0, 2).
5.Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric
problems (e.g., find the equation of a line parallel or perpendicular to a given line that
passes through a given point).
6.Find the point on a directed line segment between two given points that partitions the
segment in a given ratio.
7.Use coordinates to compute perimeters of polygons and areas of triangles and rectangles,
e.g., using the distance formula.
Explore Budgeting • 38 HIGH SCHOOL: GEOMETRY
Geometric Measurement and Dimension
STANDARD
ALIGNMENT
Explain Volume Formulas and Use Them to Solve Problems HSG-GMD.A.1 – A.3
1.Give an informal argument for the formulas for the circumference of a circle, area of a circle,
volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and
informal limit arguments.
2.Give an informal argument using Cavalieri’s principle for the formulas for the volume of a
sphere and other solid figures.
3.Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Visualize Relationships Between Two-dimensional and Three-dimensional Objects HSG-GMD.B.4
4.Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and
identify three-dimensional objects generated by rotations of two-dimensional objects.
Explore Budgeting • 39 HIGH SCHOOL: GEOMETRY
Modeling with Geometry
STANDARD
ALIGNMENT
Apply Geometric Concepts in Modeling Situations HSG-MG.A.1 – A.3
1.Use geometric shapes, their measures, and their properties to describe objects
(e.g., modeling a tree trunk or a human torso as a cylinder).
2.Apply concepts of density based on area and volume in modeling situations
(e.g., persons per square mile, BTUs per cubic foot).
3.Apply geometric methods to solve design problems (e.g., designing an object or
structure to satisfy physical constraints or minimize cost; working with typographic
grid systems based on ratios).
Explore Budgeting • 40 HIGH SCHOOL: STATISTICS AND PROBABILITY
Interpreting Categorical and Quantitative Data
STANDARD
ALIGNMENT
Summarize, Represent, and Interpret Data on a Single Count or Measurement Variable HSS-ID.A.1 – A.4
1. Represent data with plots on the real number line (dot plots, histograms, and box plots).
˜
2.Use statistics appropriate to the shape of the data distribution to compare center
(median, mean) and spread (interquartile range, standard deviation) of two or more
different data sets.
˜
3.Interpret differences in shape, center, and spread in the context of the data sets, accounting
for possible effects of extreme data points (outliers).
4.Use the mean and standard deviation of a data set to fit it to a normal distribution and
to estimate population percentages. Recognize that there are data sets for which such a
procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas
under the normal curve.
˜
Summarize, Represent, and Interpret Data on Two Categorical and Quantitative Variables HSS-ID.B.5 – B.6
5.Summarize categorical data for two categories in two-way frequency tables. Interpret
relative frequencies in the context of the data (including joint, marginal, and conditional
relative frequencies). Recognize possible associations and trends in the data.
6.Represent data on two quantitative variables on a scatter plot, and describe how the
variables are related.
a.Fit a function to the data; use functions fitted to data to solve problems in the context of
the data. Use given functions or choose a function suggested by the context. Emphasize
linear, quadratic, and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
Interpret Linear Models HSS-ID.C.7 – C.9
7.Interpret the slope (rate of change) and the intercept (constant term) of a linear model
in the context of the data.
8. Compute (using technology) and interpret the correlation coefficient of a linear fit.
9. Distinguish between correlation and causation.
Explore Budgeting • 41 HIGH SCHOOL: STATISTICS AND PROBABILITY
Making Inferences and Justifying Conclusions
STANDARD
ALIGNMENT
Understand and Evaluate Random Processes Underlying Statistical Experiments HSS-IC.A.1 – A.2
1.Understand statistics as a process for making inferences about population parameters
based on a random sample from that population.
2.Decide if a specified model is consistent with results from a given data-generating process,
e.g., using simulation. For example, a model says a spinning coin falls heads up with
probability 0.5. Would a result of 5 tails in a row cause you to question the model?
Make Inferences and Justify Conclusions from Sample Surveys, Experiments, and
Observational Studies HSS-IC.B.3 – B.6
3.Recognize the purposes of and differences among sample surveys, experiments, and
observational studies; explain how randomization relates to each.
4.Use data from a sample survey to estimate a population mean or proportion; develop
a margin of error through the use of simulation models for random sampling.
5.Use data from a randomized experiment to compare two treatments; use simulations
to decide if differences between parameters are significant.
6.Evaluate reports based on data.
Explore Budgeting • 42 HIGH SCHOOL: STATISTICS AND PROBABILITY
Conditional Probability and the Rules of Probability
STANDARD
ALIGNMENT
Understand Independence and Conditional Probability and Use Them to Interpret Data HSS-CP.A.1 – A.5
1.Describe events as subsets of a sample space (the set of outcomes) using characteristics (or
categories) of the outcomes, or as unions, intersections, or complements of other events
(“or,” “and,” “not”).
˜
2.Understand that two events A and B are independent if the probability of A and B occurring
together is the product of their probabilities, and use this characterization to determine if
they are independent.
˜
3.Understand the conditional probability of A given B as P(A and B)/P(B), and interpret
independence of A and B as saying that the conditional probability of A given B is the
same as the probability of A, and the conditional probability of B given A is the same as
the probability of B.
˜
4.Construct and interpret two-way frequency tables of data when two categories are
associated with each object being classified. Use the two-way table as a sample space to
decide if events are independent and to approximate conditional probabilities. For example,
collect data from a random sample of students in your school on their favorite subject
among math, science, and English. Estimate the probability that a randomly selected
student from your school will favor science given that the student is in tenth grade.
Do the same for other subjects and compare the results.
˜
5.Recognize and explain the concepts of conditional probability and independence in
everyday language and everyday situations. For example, compare the chance of having
lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
Use the Rules of Probability to Compute Probabilities of Compound Events HSS-CP.B.6 – B.9
6.Find the conditional probability of A given B as the fraction of B’s outcomes that also belong
to A, and interpret the answer in terms of the model.
7.Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in
terms of the model.
8.Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)
P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
9.Use permutations and combinations to compute probabilities of compound events and
solve problems.
Explore Budgeting • 43 HIGH SCHOOL: STATISTICS AND PROBABILITY
Using Probability to Make Decisions
STANDARD
ALIGNMENT
Calculate Expected Values and Use Them to Solve Problems HSS-MD.A.1 – A.4
1.Define a random variable for a quantity of interest by assigning a numerical value to each
event in a sample space; graph the corresponding probability distribution using the same
graphical displays as for data distributions.
2.Calculate the expected value of a random variable; interpret it as the mean of the
probability distribution.
3.Develop a probability distribution for a random variable defined for a sample space in which
theoretical probabilities can be calculated; find the expected value. For example, find the
theoretical probability distribution for the number of correct answers obtained by guessing
on all five questions of a multiple-choice test where each question has four choices, and find
the expected grade under various grading schemes.
4.Develop a probability distribution for a random variable defined for a sample space in which
probabilities are assigned empirically; find the expected value. For example, find a current
data distribution on the number of TV sets per household in the United States, and calculate
the expected number of sets per household. How many TV sets would you expect to find in
100 randomly selected households?
Use Probability to Evaluate Outcomes of Decisions HSS-MD.B.5 – B.7
5.Weigh the possible outcomes of a decision by assigning probabilities to payoff values
and finding expected values.
a.Find the expected payoff for a game of chance. For example, find the expected winnings
from a state lottery ticket or a game at a fast-food restaurant.
b.Evaluate and compare strategies on the basis of expected values. For example, compare
a high-deductible versus a low-deductible automobile insurance policy using various,
but reasonable, chances of having a minor or a major accident.
6.Use probabilities to make fair decisions (e.g., drawing by lots, using a random
number generator).
7.Analyze decisions and strategies using probability concepts (e.g., product testing,
medical testing, pulling a hockey goalie at the end of a game).
Explore Budgeting • 44